A394445 - OEIS

%I #27 May 01 2026 00:36:43

%S 9,9,15,20,25,31,49,41,54,61,77,85,104,113,135,145,170,181,209,221,

%T 252,265,299,313,350,365,405,421,464,481,527,545,594,613,665,685,740,

%U 761,819,841,902,925,989

%N a(n) is the minimum k such that every 2-coloring of {1, ..., k} contains a monochromatic solution to x + y = n*z with x, y, z pairwise distinct.

%C These are "distinct-variable" 2-color Rado numbers for the equation x + y = n*z. The standard (non-distinct) 2-color Rado numbers, where variables may repeat, are n*(n+1)/2 for n >= 4 (Burr and Loo, see A100542). For n=1 this is the Schur equation x+y=z with distinct variables, giving the weak Schur number WS(2)+1 = 9 (cf. A072842). For n=2 this is x+y=2*z with distinct variables, equivalent to 3-term arithmetic progressions, giving the van der Waerden number W(3;2) = 9. The sequence is not monotone: a(7) = 49 > a(8) = 41.

%D S. A. Burr and S. P. Loo, On Rado numbers II, unpublished manuscript, c. 1992.

%D H. Harborth and S. Maasberg, All two-color Rado numbers for a(x+y)=bz, Discrete Math., 197/198 (1999), 397-407.

%D S. Jones and D. Schaal, Two-color Rado numbers for x+y+c=kz, Discrete Math., 289 (2004), 63-69.

%H Alex Towell, <a href="/A394445/b394445.txt">Table of n, a(n) for n = 1..500</a>

%H Alex Towell, <a href="https://zenodo.org/record/19372727">Rado Numbers for x+y=kz: Computation, Closed-Form Formula, and Complete Proof</a>, Zenodo, 2026.

%F For n >= 8, a(n) = n*(n+3)/2 if n is odd, a(n) = (n^2 + 2*n + 2)/2 if n is even.

%e a(3) = 15 because every 2-coloring of {1, ..., 15} contains a monochromatic solution to x+y=3z with x, y, z distinct, but the 2-coloring {2, 5, 8, 9, 11, 12, 14} red, {1, 3, 4, 6, 7, 10, 13} blue of {1, ..., 14} avoids all monochromatic solutions.

%o (Python)

%o from pysat.solvers import Solver

%o def a(n, max_N=2000):

%o     for N in range(1, max_N):

%o         sols = []

%o         for z in range(1, N+1):

%o             for x in range(1, N+1):

%o                 y = n*z - x

%o                 if 1 <= y <= N and len({x, y, z}) == 3:

%o                     sols.append((x, y, z))

%o         with Solver(name="cd195") as s:

%o             for i in range(1, N+1):

%o                 s.add_clause([2*i-1, 2*i])

%o                 s.add_clause([-(2*i-1), -(2*i)])

%o             for x, y, z in sols:

%o                 for v in [lambda i: 2*i-1, lambda i: 2*i]:

%o                     s.add_clause([-v(x), -v(y), -v(z)])

%o             if not s.solve():

%o                 return N

%Y Cf. A100542 (non-distinct variant), A072842 (weak Schur numbers), A003071 (Schur numbers).

%Y Cf. A000096, A001844.

%K nonn

%O 1,1

%A _Alex Towell_, Mar 20 2026